How can I make my network treat rotations of the input equally?
If I understand well your single output node will be the next status of the square in the middle. You don't need to worry about the number of nodes in the hidden layers while you have sufficient resources to train the model. This problem is very easy to learn for a neural network so no size concerns.

You need to do supervised training that means you need to feed in input data and the matching expected output. You need to be sure that in your training data all 4 rotations are assigned to the same output. This way your network should learn to treat all of these the same way.  

You made me curious so I tried myself. My solution could learn 100% correct in about 20 epochs running within a few seconds on my old laptop. I only slightly changed the output to be categorical either [0,1] or [1,0] but that gives the same result that you are looking for. Just for reference here is the code written in python:  

from keras.models import Sequential
from keras.layers import Input, Dense
from keras.models import Model
from keras import optimizers
from keras.utils.np_utils import to_categorical
import helper

x_,y_ = helper.fnn_csv_toXY("conway.csv","output",False)
y_binary = to_categorical(y_)

model = Sequential()
model.add(Dense(100, activation='relu', kernel_initializer='glorot_uniform',input_shape =(9,)))
model.add(Dense(20, activation='relu', kernel_initializer='glorot_uniform'))
model.add(Dense(2, activation='softmax'))
adam=optimizers.Adam()
model.compile(optimizer=adam,
              loss='categorical_crossentropy',
              metrics=['acc'])
model.fit(x_, y_binary, epochs=100)